In an $A.P.$,the sixth term is $a_6 = 2$. If the product $a_1 a_4 a_5$ is the greatest,then the common difference of the $A.P.$ is equal to

If $a, b, c$ are in $A.P.$,then $(a + 2b - c)(2b + c - a)(c + a - b)$ equals

Let $AP(a; d)$ denote the set of all the terms of an infinite arithmetic progression with first term $a$ and common difference $d > 0$. If $AP(1; 3) \cap AP(2; 5) \cap AP(3; 7) = AP(a; d)$,then $a + d$ equals:

If the sum of $n$ terms of an $A$.$P$. is $3n^2 + 5n$ and its $m$th term is $164$,then the value of $m$ is

After inserting $n$ arithmetic means $(A.M.s)$ between $2$ and $38$,the sum of the resulting progression is $200$. The value of $n$ is

Consider an $A$.$P$.: $a_1, a_2, \dots, a_n$,with $a_1 > 0$. If $a_2 - a_1 = -\frac{3}{4}$,$a_n = \frac{1}{4} a_1$,and $\sum_{i=1}^n a_i = \frac{525}{2}$,then $\sum_{i=1}^{17} a_i$ is equal to: